which is usually imposed when working with \(Z_n\)  (see e.g. [1]}, [2]}).
Due to the denominator \(V_n\) , the characteristic function of \(T_n\)  lacks the typical product form, required for an Edgeworth expansion. To overcome this problem, we first condition on \(\mathcal {F}_n=\sigma (|X_1|,\dots ,|X_n|)\)  as in [3]}. On the one hand, the independence of the factors then provides a desired structure as a product. On the other hand, the random variables \(X_1,\dots ,X_n\)  conditional on \(\mathcal {F}_n\)  are discrete (non-lattice) random variables that are not identically distributed any longer. Therefore, the necessary Cramér condition eq.cramer is not fulfilled and the standard theorems for Edgeworth expansions (such as from [1]}, [2]}) do not apply. These properties necessitate adjustments in the derivation of an Edgeworth expansion, but lead to rather involved integral remainder terms which have to be bounded thoroughly. At this point, the non-singularity is getting essential.
